Tessellations of rational complex functions and the Riemann's existence theorem

TL;DR

This work links tessellations on compact Riemann surfaces to complex rational functions through the Schwarz-Klein algorithm, which assigns to a function $R$ of degree $n\ge 2$ a tessellation with $2n$ tiles, each a topological $q$-gon, using a cyclic order of the $q$ critical values. It proves a converse: given a tessellation with a consistent $q$-labelling meeting genus and gonality bounds, there exists a (generally nonunique) triple $(M,R,\mathcal{L}_\gamma)$ whose tessellation $\mathscr{T}_\gamma(R)$ matches the given one up to orientation-preserving equivalence, thereby realizing the Riemann existence theorem in a combinatorial framework. The paper develops precise correspondences between analytic data and combinatorial structures via edge subdivisions and a surgery-based gluing process, enabling construction of $R$ and the complex structure from tessellations. The framework is illustrated with diverse examples, including non-generic tessellations, multiple admissible $q$-labellings, fortunate rational functions, and classical cases like Weierstrass functions and hyperelliptic surfaces, highlighting potential connections to dessins d'enfants and differential equations.

Abstract

A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T.

Figures (7)

• Figure 1: Affine view of the tessellation of a non generic rational function $R(z)$ of degree 5. a) The ${\tt t}$--graph $\Gamma= R^{-1} ({\mathbb R} \cup \{ \infty\})$ and its non homogeneous tessellation $\mathscr{T}(\Gamma)$. b) The consistent $6$--labelling $\mathcal{L}_c: V(\Gamma) \longrightarrow {\mathbb Z}_6$, where $\mathcal{L}(\infty)= 6$. c) The ${\tt R}$--map $\widehat{\Gamma}= R^* \gamma$ and its homogeneous tessellation $\mathscr{T}(\widehat{\Gamma})$: it has a consistent $6$--labelling $R^*\mathcal{L}_c$, each tile is a $6$--gon, with vertices at the (red) critical points $\mathcal{CP}_R$, the point $w_6=\infty$ (which has label 6), and the (green) cocritical points $\mathcal{C}c_R$.
• Figure 2: Consistent $q$--labellings for $q=4$, and $5$ of the same ${\tt t}$--graph of Figure \ref{['fig:tres-mosaicos-etiquetas-h']}.a which originates from $R(z)$ in Example \ref{['ex:racional-cocriticos']}. a) A consistent $4$--labelling $\mathcal{L}_c: V(\Gamma) \longrightarrow {\mathbb Z}_4$, where $\mathcal{L}(\infty)= 4$. b) A consistent $5$--labelling $\mathcal{L}_c: V(\Gamma) \longrightarrow {\mathbb Z}_5$, where $\mathcal{L}(\infty)= 5$. c) A topologically different consistent $5$--labelling $\mathcal{L}_c: V(\Gamma) \longrightarrow {\mathbb Z}_5$, where $\mathcal{L}(\infty)= 5$. Each case produces a different ${\tt R}$--map $\widehat{\Gamma}= R^* \gamma$ with a homogeneous tessellation $\mathscr{T}(\widehat{\Gamma})$. Each tile is a $q$--gon, with vertices at the (red) critical points $\mathcal{CP}_R$, the point $\infty$ with label $q$, and the (green) cocritical points $\mathcal{C}c_R$.
• Figure 3: Two topological $q$--gons $T_1, \,T_{1}'$ are mapped to the half planes ${\mathbb H}^{2} _{+}, \, {\mathbb H}^{2} _{-}$. We sketch the case $q=7$, the critical points are red points in $\mathcal{M}$, a cocritical point is colored green.
• Figure 4: Zoom of Figure \ref{['fig:tres-mosaicos-etiquetas-h']} illustrates four tiles before the gluing process. The tiles are provided with the real trajectories of the vector fields $\Psi_{\alpha \pm}^* \frac{\partial }{\partial x}$ in black and $\Psi_{\alpha \pm}^* \frac{\partial }{\partial y}$ in red. Moreover, the critical points are red, the cocritical points are green. A simple pole of $R(z)$ appears as cocritical point.
• Figure 5: Schwarz's tessellations. The left column illustrates the $k$--dihedron and the platonic solids. The other columns show tessellations of rational $G$--invariant Belyı functions.

Theorems & Definitions (21)

• Theorem 1
• Definition 1
• Definition 2
• Definition 2
• Definition 3
• Remark 1
• Example 1
• Definition 4
• Remark 2: Consistent $q$--labelling for ${\tt R}$--maps
• Example 2: Tessellation of a non generic rational function